Lemma 4.5 
[1]}
Let \(\lbrace \alpha _n\rbrace _{n=1}^\infty \)  be a sequence converging to infinity. Let \(\mu _n\) , \(\sigma ^2_n\) , and \(f_n\)  denote the mean, variance and density of \(\mathrm {Beta}(\alpha _n,n-\alpha _n+1)\) , respectively. Let \(g(n)\)  be any positive function satisfying \(g(n) = o(\min \lbrace \alpha _n, n-\alpha _n\rbrace )\)  as \(n\rightarrow \infty \) . Then,
\(f_n(\mu _n + \sigma _n \cdot t) \ge \frac{e^{-t^2/2}}{\sqrt{2\pi } \cdot \sigma _n } \left( 1 - \frac{t^3}{\sqrt{g(n)}} - \frac{1}{g(n)} \right)\) 

